Let $S$ and $A$ be countable sets and let $\mathscr{G}(\Pi)$ be the set of Markov random fields on $S^A$ (with the $\sigma$-field generated by the finite cylinder sets) corresponding to a specification $\Pi$, Markov with respect to a tree-like neighbour relation in $A$. We define the class $\mathscr{M}(\Pi)$ of Markov chains in $\mathscr{G}(\Pi)$, and generalise results of Spitzer to show that every extreme point of $\mathscr{G}(\Pi)$ belongs to $\mathscr{M}(\Pi)$. We establish a one-to-one correspondence between $\mathscr{M}(\Pi)$ and a set of "entrance laws" associated with $\Pi$. These results are applied to homogeneous Markov specifications on regular infinite trees. In particular for the case $|S| = 2$ we obtain a quick derivation of Spitzer's necessary and sufficient condition for $|\mathscr{G}(\Pi)| = 1$, and further show that if $|\mathscr{M}(\Pi)| > 1$ then $|\mathscr{M}(\Pi)| = \infty$.
Publié le : 1983-11-14
Classification:
Phase transition,
Markov random fields,
Markov chains on infinite trees,
entrance laws,
60G60,
60K35,
82A25,
60J10
@article{1176993439,
author = {Zachary, Stan},
title = {Countable State Space Markov Random Fields and Markov Chains on Trees},
journal = {Ann. Probab.},
volume = {11},
number = {4},
year = {1983},
pages = { 894-903},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993439}
}
Zachary, Stan. Countable State Space Markov Random Fields and Markov Chains on Trees. Ann. Probab., Tome 11 (1983) no. 4, pp. 894-903. http://gdmltest.u-ga.fr/item/1176993439/